Chua's Oscillator in Musical Applications

Using the sounds of chaos in music and sound design

For
decades, musicians have been experimenting and working with electronic
synthesizers to generate sound. In some cases the reasoning is logistical
- often it is prohibitively expensive, for example, to haul a heavy
and expensive piano or organ to a venue. Many times, however, the goal
is creative expression in the generation of new and previously unheard
or unharnessed sounds. Chaotic oscillators, including the example discovered
by Leon Chua, open the door to a new universe of sonic possibilities.

Chua's
oscillator is a system described by a set of three differential equations
that can be realized either in digital form or in analog form using
opamps and passive circuit components, simple in appearance, but extraordinarily
complex in its analysis and behavior. In the video,
you are listening to two of these three signals, at first from Chua's
circuit acting alone, and later from this same circuit used similarly
to a module in an analog synthesizer, acting through effects and with
varying levels of this effected signal feeding back to further perturb
the behavior of the oscillator. The video is a phase plot of the two
dry (non-effected) signals from the circuit (one moving the display
beam left and right, and the other up and down) on an analog oscilloscope.

The
sounds from Chua's circuit are widely varied, ranging from pure sine
waves to almost pure noise, with many varied behaviors within. Period
doubling and intermittency effects can be particularly useful from a
sonic standpoint, especially when processed through appropriate effects,
such as delays, reverbs, resonant filters, ring modulators, and pitch
shifters. A suitable control interface for the many parameters of the
oscillator and effects system is necessary for full realization of the
sonic potential of chaos.

This system found extensive use in a sound design for Marisol
at Southern Illinois University at Edwardsville and was received with
very positive responses from critics, audiences, cast, and crew alike.
The eerie sounds generated by the combinations of chaotic oscillator
with effects fit the bill perfectly to accent the dark, apocalyptic
tone of the play, but certain ranges of the system's behavior and combinations
of effects, and possibly even by utilizing different chaotic systems,
can produce sounds with a wide and emotionally expressive variety.

Future
opportunities in this area of research include mapping the system's
parameter-space behavior (the complexity of
which is astounding - consisting of fractal-like clouds of points where
various behaviors take place), using this information to develop new
methods and interfaces for control, the use of gyrator circuits to lessen
the expense and simplify control of the circuit's reactive elements,
investigating the effects of various feedback loop effects topologies,
the use of multiple interacting systems to create further sonic variety,
implementation in digital form (which could reduce cost and potentially
simplify many control problems, likely at the cost of fine nuance in
the behavior), and similar investigation of or exploration for other
suitable chaotic oscillator systems.

Below
are a few pictures of the phase plots of the Chua circuit's behavior.
Click on the picture to hear an audio sample of the sounds of chaos.
Note how at the end of the series, the stable attractors that appear
as the variable inductor (pictured at right with circuit board) is increased
decrease in pitch roughly with the musical scale. This is inherent to
the behavior of the system - these four stable tones are like islands
in a sea of chaos as the value of the inductor is changed - and while
these islands don't always align the intervals with our musical scale,
behaviors like these could still be valuable in harnessing the system
musically.

Photos (click image to hear mp3 sound clip):

The next seven pictures and sound clips illustrate the musical tune
of the Chua's stable orbits as the inductance is decreased, and the
chaotic behavior between these orbits. The tuner software has been calbrated
to F3 = 92Hz, but I believe that the base frequency of this behavior
could be tuned by changing the capacitances of the circuit.

Parameter-space behavior:

The plots below are from a MATLAB simulation of the chua oscillator
system. X and Y directions each correspond to varying a particular parameter
of the system and the Z direction (represented by height and/or color)
is a calculated value that reflects the level of chaos in the waveform
by measuring the spread of the frequency distribution of the signal
using an FFT and additional mathematical processing of the frequency
spectrum..

Higher values of Z (brighter colors and/or higher altitude) indicate
that the output power is spread among more frequencies, which is indicative
of the onset of chaos and corresponds to more harmonics and/or more
chaotic noise in terms of audio output. Lower values of Z indicate simpler
tones, to a minimum of 1 when the signal is at a single fundamental
frequency or 0 when there is no AC signal, as in the case when the system
comes to rest at a steady-state (DC) solution.

The graphs act as a map to the oscillator's behavior. Within a simulator,
or with the parameters scaled to real-world component values in a hardware
implementation, they should allow one to pick a range of behavior and
find combinations of parameters that fulfill the requirements.